Finite Group Databasebeta

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Cayley diagram

C2×#7858 #180866

order512
nilpotency class2
isoclinismisoclinic class of #7855
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42

Aliases

direct productC2×#7858

Subgroups

subgroupG/H
centerC22×C4C2×C42
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,11,13,17)(6,9)(7,10)(8,12,15,18)(14,16)(19,20,23,27,24,28,22,26)(21,25,29,32,30,33,31,34);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)(19,21)(20,25)(22,31)(23,29)(24,30)(26,34)(27,32)(28,33);(19,22,24,23)(20,26,28,27)(21,29,30,31)(25,32,33,34)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303344, C2×#233571, #72303381
maximal normal subgroup of#72303344, C2×#233571, #72303381

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88062);